3.2846 \(\int \frac {1}{\sqrt {1-x} \sqrt {3-x} \sqrt {2+x}} \, dx\)

Optimal. Leaf size=25 \[ \frac {2 \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {x+2}}{\sqrt {3}}\right ),\frac {3}{5}\right )}{\sqrt {5}} \]

[Out]

2/5*EllipticF(1/3*(2+x)^(1/2)*3^(1/2),1/5*15^(1/2))*5^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {119} \[ \frac {2 F\left (\sin ^{-1}\left (\frac {\sqrt {x+2}}{\sqrt {3}}\right )|\frac {3}{5}\right )}{\sqrt {5}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - x]*Sqrt[3 - x]*Sqrt[2 + x]),x]

[Out]

(2*EllipticF[ArcSin[Sqrt[2 + x]/Sqrt[3]], 3/5])/Sqrt[5]

Rule 119

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(b/d)] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-(d/b), 0]) &&  !(SimplerQ[c +
d*x, a + b*x] && GtQ[(-(b*e) + a*f)/f, 0] && GtQ[-(f/b), 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[(-(d*e)
+ c*f)/f, 0] && GtQ[(-(b*e) + a*f)/f, 0] && (PosQ[-(f/d)] || PosQ[-(f/b)]))

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-x} \sqrt {3-x} \sqrt {2+x}} \, dx &=\frac {2 F\left (\sin ^{-1}\left (\frac {\sqrt {2+x}}{\sqrt {3}}\right )|\frac {3}{5}\right )}{\sqrt {5}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.09, size = 68, normalized size = 2.72 \[ -\frac {2 \sqrt {\frac {x-3}{x-1}} (x-1) \sqrt {\frac {x+2}{x-1}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {3}}{\sqrt {1-x}}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {-x^2+x+6}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 - x]*Sqrt[3 - x]*Sqrt[2 + x]),x]

[Out]

(-2*Sqrt[(-3 + x)/(-1 + x)]*(-1 + x)*Sqrt[(2 + x)/(-1 + x)]*EllipticF[ArcSin[Sqrt[3]/Sqrt[1 - x]], -2/3])/(Sqr
t[3]*Sqrt[6 + x - x^2])

________________________________________________________________________________________

fricas [F]  time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x + 2} \sqrt {-x + 3} \sqrt {-x + 1}}{x^{3} - 2 \, x^{2} - 5 \, x + 6}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/2)/(3-x)^(1/2)/(2+x)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(x + 2)*sqrt(-x + 3)*sqrt(-x + 1)/(x^3 - 2*x^2 - 5*x + 6), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x + 2} \sqrt {-x + 3} \sqrt {-x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/2)/(3-x)^(1/2)/(2+x)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x + 2)*sqrt(-x + 3)*sqrt(-x + 1)), x)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 21, normalized size = 0.84 \[ \frac {2 \sqrt {3}\, \EllipticF \left (\frac {\sqrt {5 x +10}}{5}, \frac {\sqrt {15}}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x+1)^(1/2)/(-x+3)^(1/2)/(x+2)^(1/2),x)

[Out]

2/3*EllipticF(1/5*(5*x+10)^(1/2),1/3*15^(1/2))*3^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x + 2} \sqrt {-x + 3} \sqrt {-x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/2)/(3-x)^(1/2)/(2+x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x + 2)*sqrt(-x + 3)*sqrt(-x + 1)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\sqrt {1-x}\,\sqrt {x+2}\,\sqrt {3-x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x)^(1/2)*(x + 2)^(1/2)*(3 - x)^(1/2)),x)

[Out]

int(1/((1 - x)^(1/2)*(x + 2)^(1/2)*(3 - x)^(1/2)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {1 - x} \sqrt {3 - x} \sqrt {x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(1/2)/(3-x)**(1/2)/(2+x)**(1/2),x)

[Out]

Integral(1/(sqrt(1 - x)*sqrt(3 - x)*sqrt(x + 2)), x)

________________________________________________________________________________________